It doesn't matter much which pet you have. Some come with higher stats & more runes, but you can build anything up to be just as good. The more expensive/valuable the pet, the more it'll come with. If your lava snail has 20 good runes and 100 stats, getting a new one won't help. Consider swapping out cheap charms for expensive charms. If you want to play with stats, it may be easier to start over, or it may be easier to get a bunch of -3 stat items to get growth capacity back, then +3 stat items to refocus.

I wonder if they group battles & races by level. Level 30 players will probably have better setup pets than level 10 players, and I think the groups are segregated. Most of my opponents now are level 40+, but the battle I just did on my level 12 guy, he was up against a lot in the 10s and 20s. You may have just eeked into the big leagues, so your opponents have better pets now, and that's the difference. Check the level of your competitors.

Also about pet fights, there is a popular rumour that the pet stats correspond to character stats - just match the colour - and they do better with runes which represent the weapon types that are based on the stats they have in plenty. For example that a pet with 100 cruelty would be really good with shadow, axes and fire...etc Just a rumour tho